Question: Solve for $x$ : $ 7|x - 10| - 9 = 6|x - 10| + 7 $
Subtract $ {6|x - 10|} $ from both sides: $ \begin{eqnarray} 7|x - 10| - 9 &=& 6|x - 10| + 7 \\ \\ { - 6|x - 10|} && { - 6|x - 10|} \\ \\ 1|x - 10| - 9 &=& 7 \end{eqnarray} $ Add ${9}$ to both sides: $ \begin{eqnarray} 1|x - 10| - 9 &=& 7 \\ \\ { + 9} &=& { + 9} \\ \\ 1|x - 10| &=& 16 \end{eqnarray} $ Simplify: $ |x - 10| = 16$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 10 = -16 $ or $ x - 10 = 16 $ Solve for the solution where $x - 10$ is negative: $ x - 10 = -16 $ Add ${10}$ to both sides: $ \begin{eqnarray} x - 10 &=& -16 \\ \\ {+ 10} && {+ 10} \\ \\ x &=& -16 + 10 \end{eqnarray} $ $ x = -6 $ Then calculate the solution where $x - 10$ is positive: $ x - 10 = 16 $ Add ${10}$ to both sides: $ \begin{eqnarray} x - 10 &=& 16 \\ \\ {+ 10} && {+ 10} \\ \\ x &=& 16 + 10 \end{eqnarray} $ $ x = 26 $ Thus, the correct answer is $x = -6 $ or $x = 26 $.